The finite element method is a powerful tool for calculating stress in complicated shell and plate structures that are difficult to analyze by classical plate and shell theories. The method consists of subdividing a given domain into small elements connected at the nodal points as shown in Fig. 16-1. The mathematical formulation consists of combining the governing equations of each of the elements to form a solution for the domain that satisfies the boundary conditions. The approximations associated with finite element solutions depend on many variables such as the type of element selected, number of elements used to model the domain, and the boundary conditions.
The complete derivation of the various equations for one-, two- and three-dimensional elements is beyond the scope of this book. However, a few equations are derived here to demonstrate the basic concept of Finite Element formulation and its applicability to the solution of plates and shells.
We begin the derivations by defining various elements (Fig. 16-2) and terms. Figure 16-2a shows a one-dimensional element in the x-direction with two nodal points, i and j. Figure 16-2b shows a two-dimensional triangular element in the x, y plane with nodal points i, j, and k. And Fig. 16-2c shows a three-dimensional rectangular brick element with eight nodal points.